On Tate-Shafarevich groups over galois extensions.

Let A be an abelian variety defined over a number field K. Let L be a finite Galois extension of K with Galois group G and let III( A/K) and III( A/L) denote, respectively, the Tate-Shafarevich groups of A over K and of A over L. Assuming these groups are finite, we compute [III( A/L) ^G ]/[III( A/K)] and [III( A/K)]/[ N(III( A/L))], where [ X] is the order of a finite abelian group X. Especially, when L is a quadratic extension of K, we derive a simple formula relating [III( A/L)], [III( A/K)], and [III( A ^x/ K)] where A x is the twist of A by the non-trivial character χ of G. [ABSTRACT FROM AUTHOR]